Let F be a collection of disjoint intervals of real numbers. This means that if I and J are two distinct intervals in F, then the intersection of I and J is zero. Prove that the set F is countable.
Between any two real numbers there is a rational number.
Properly defined a closed interval looks like $\displaystyle \left[ {a,b} \right] = \left\{ {x:a < b\;\& \;a \leqslant x \leqslant b} \right\}$.
Thus in each closed interval in the collection $\displaystyle \mathcal{F}$ there is a rational number.
Because the set of rationals is countable the collection $\displaystyle \mathcal{F}$ is countable.